Affine buildings for dihedral groups

We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of affine buildings by constructing a new class of rank‑2, thick, nondiscrete affine buildings that are associated with any finite dihedral group Dₙ (n ≥ 2). Traditional affine buildings are usually discrete: their apartments are modeled on Euclidean tilings by regular polygons with a lattice‑type metric, and the Weyl group acts by a finite reflection group on a discrete set of chambers. In contrast, the authors develop a framework in which the apartments are still Euclidean planes tiled by regular 2n‑gons, but the metric is continuous, allowing distances to take arbitrary real values. This shift from a lattice to a real‑valued metric is what makes the buildings nondiscrete.

The construction proceeds in several stages. First, the authors review the algebraic structure of the dihedral group Dₙ, emphasizing its presentation as a group generated by two reflections whose product is a rotation of order n. They then describe how Dₙ acts on the Euclidean plane by reflecting across the axes that bound a regular 2n‑gon. By repeatedly applying these reflections, one obtains an infinite tiling of the plane by congruent 2n‑gons; each edge of a tile is identified as a wall, and each tile itself is a chamber. This tiling serves as the model for a single apartment.

To achieve thickness, the authors introduce an infinite family of parallel copies of the basic tiling, each shifted by a continuous parameter along the direction orthogonal to a chosen wall. Because the shift is not restricted to integer multiples of a fixed length, every wall is intersected by infinitely many apartments, guaranteeing that each wall is contained in at least two distinct chambers. The resulting structure satisfies the “thick” condition of affine building theory, which requires that every codimension‑1 face (wall) be adjacent to at least three chambers in higher rank, and at least two in rank 2.

The paper then verifies the key axioms of an affine building for the constructed space. Strong transitivity is proved by showing that any two apartments can be mapped onto each other by an element of the Weyl group Dₙ combined with an appropriate translation in the continuous parameter. Regularity follows because every apartment is isometric to the standard 2n‑gon tiling, and the action of Dₙ is cocompact on the set of apartments. The authors also demonstrate that the metric space is complete: Cauchy sequences converge because the underlying Euclidean plane is complete, and the continuous shift parameter does not introduce any pathological gaps.

A substantial portion of the work is devoted to comparing the new nondiscrete buildings with classical discrete affine buildings. The authors point out that while the combinatorial incidence structure (chambers, walls, apartments) remains formally similar, the geometry is markedly different. In the nondiscrete case, the visual boundary of the building is a circle equipped with a natural Dₙ‑invariant topology, whereas in the discrete case the boundary is a totally disconnected set. Moreover, the continuous metric allows for a richer family of geodesics and a more flexible notion of curvature, opening potential connections to CAT(0) geometry and to the theory of metric spaces with group actions.

Finally, the paper discusses implications and future directions. The construction shows that any finite dihedral group can serve as the Weyl group of a thick, nondiscrete affine building, suggesting that similar methods might be applied to other finite reflection groups, possibly yielding new families of buildings that are not captured by the classical Bruhat–Tits theory. The authors also propose investigating the representation theory of groups acting on these buildings, exploring harmonic analysis on the associated boundaries, and extending the construction to higher rank situations where the Weyl group is a product of dihedral groups. In summary, the work provides a concrete, fully worked‑out example of a nondiscrete affine building, broadening the landscape of geometric group theory and offering a fertile ground for further research.